ONT Re: Information = Comprehension x Extension -- Discussion
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ICE Discussion. Note 31
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Re: ICE Discussion. http://suo.ieee.org/ontology/thrd1.html#05274
NB. off list questions from a couple of readers
prompt me to send the following information
about the context of the example, plus what
happened to some of the internal linkages.
this example comes from c.s. peirce's lectures at harvard
and the lowell institute in 1865-66 where he introduced
his "theory of information".
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures of 1865, pages 161-302 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
here are excerpts for the present context:
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ICE. Note 18
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| We come next to consider inductions. In inferences of this kind
| we proceed as if upon the principle that as is a sample of a class
| so is the whole class. The word 'class' in this connection means
| nothing more than what is denoted by one term, -- or in other words
| the sphere of a term. Whatever characters belong to the whole sphere
| of a term constitute the content of that term. Hence the principle of
| induction is that whatever can be predicated of a specimen of the sphere
| of a term is part of the content of that term. And what is a specimen?
| It is something taken from a class or the sphere of a term, at random --
| that is, not upon any further principle, not selected from a part of
| that sphere; in other words it is something taken from the sphere
| of a term and not taken as belonging to a narrower sphere. Hence
| the principle of induction is that whatever can be predicated of
| something taken as belonging to the sphere of a term is part of
| the content of that term. But this principle is not axiomatic
| by any means. Why then do we adopt it?
|
| CSP, CE 1, pages 462-463.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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ICE. Note 19
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| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge; it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference. Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term; and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals. But a little reflection will show that when our
| knowledge receives an addition this principle does not hold.
|
| Thus suppose a blind man to be told that no red things are
| blue. He has previously known only that red is a color;
| and that certain things 'A', 'B', and 'C' are red.
|
| The comprehension of red then has been for him 'color'.
| Its extension has been 'A', 'B', 'C'.
|
| But when he learns that no red thing is blue, 'non-blue'
| is added to the comprehension of red, without the least
| diminution of its extension.
|
| Its comprehension becomes 'non-blue color'.
| Its extension remains 'A', 'B', 'C'.
|
| Suppose afterwards he learns that a fourth thing 'D' is red.
| Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'. Thus, the rule
| that the greater the extension of a term the less its comprehension
| and 'vice versa', holds good only so long as our knowledge is not
| added to; but as soon as our knowledge is increased, either the
| comprehension or extension of that term which the new information
| concerns is increased without a corresponding decrease of the other
| quantity.
|
| The reason why this takes place is worthy of notice. Every addition to
| the information which is incased in a term, results in making some term
| equivalent to that term. Thus when the blind man learns that 'red' is
| not-blue, 'red not-blue' becomes for him equivalent to 'red'. Before
| that, he might have thought that 'red not-blue' was a little more
| restricted term than 'red', and therefore it was so to him, but
| the new information makes it the exact equivalent of red.
| In the same way, when he learns that 'D' is red, the
| term 'D-like red' becomes equivalent to 'red'.
|
| Thus, every addition to our information about a term is an addition
| to the number of equivalents which that term has. Now, in whatever
| way a term gets to have a new equivalent, whether by an increase in
| our knowledge, or by a change in the things it denotes, this always
| results in an increase either of extension or comprehension without
| a corresponding decrease in the other quantity.
|
| For example we have here a number of circles
| dotted and undotted, crossed and uncrossed:
|
| (·X·) (···) (·X·) (···) ( X ) ( ) ( X ) ( )
|
| Here it is evident that the greater the extension the
| less the comprehension:
|
| o-------------------o-------------------o
| | | |
| | dotted | 4 circles |
| | | |
| o-------------------o-------------------o
| | | |
| | dotted & crossed | 2 circles |
| | | |
| o-------------------o-------------------o
|
| Now suppose we make these two terms 'dotted circle'
| and 'crossed and dotted circle' equivalent. This we can
| do by crossing our uncrossed dotted circles. In that way,
| we increase the comprehension of 'dotted circle' and at the
| same time increase the extension of 'crossed and dotted circle'
| since we now make it denote 'all dotted circles'.
|
| CSP, CE 1, pages 463-464.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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ICE. Note 20
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| Thus every increase in the number of equivalents of any term increases either
| its extension or comprehension and 'conversely'. It may be said that there
| are no equivalent terms in logic, since the only difference between such
| terms would be merely external and grammatical, while in logic terms
| which have the same meaning are identical. I fully admit that.
| Indeed, the process of getting an equivalent for a term is
| an identification of two terms previously diverse. It is,
| in fact, the process of nutrition of terms by which they
| get all their life and vigor and by which they put forth
| an energy almost creative -- since it has the effect of
| reducing the chaos of ignorance to the cosmos of science.
| Each of these equivalents is the explication of what there is
| wrapt up in the primary -- they are the surrogates, the interpreters
| of the original term. They are new bodies, animated by that same soul.
| I call them the 'interpretants' of the term. And the quantity of these
| 'interpretants', I term the 'information' or 'implication' of the term.
|
| CSP, CE 1, pages 464-465.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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ICE. Note 21
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| We must therefore modify the law of
| the inverse proportionality of
| extension and comprehension
| and instead of writing
|
| Extension x Comprehension = Constant
|
| which crudely expresses the fact
| that the greater the extension the
| less the comprehension, we must write
|
| Extension x Comprehension = Information
|
| which means that when the information
| is increased there is an increase of
| either extension or comprehension
| without any diminution of the
| other of these quantities.
|
| Now, ladies and gentlemen, as it is true that
| every increase of our knowledge is an increase
| in the information of a term -- that is, is an
| addition to the number of terms equivalent to
| that term -- so it is also true that the first
| step in the knowledge of a thing, the first
| framing of a term, is also the origin of the
| information of that term because it gives the
| first term equivalent to that term. I here
| announce the great and fundamental secret
| of the logic of science. There is no term,
| properly so called, which is entirely destitute
| of information, of equivalent terms. The moment
| an expression acquires sufficient comprehension
| to determine its extension, it already has more
| than enough to do so.
|
| CSP, CE 1, page 465.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
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the database that my e-friend elijah wright built for the inquiry list
got a bit scrambled yesterday and i don't know if it will get scrambled
back anytime soon. so quite a few of the urls that i gave in old posts
have now been shifted. should they stabilize in their present locations
i will eventually revise my local indexes and supply a new concordance
at some point. working on the philosophy that what does not kill us
we should write papers about, i will then initiate an inquiry into
the "finite difference calculus of urls and the effects of random
shift operators thereon".
the thread head for the most recent discussion on the ontology list
is given above, and the first note there refers to the older set of
excerpts on the inquiry list, that used to be here:
ICE. http://stderr.org/pipermail/inquiry/2003-March/thread.html#194
ICE. http://stderr.org/pipermail/inquiry/2003-April/thread.html#356
but now they are here:
ICE. http://stderr.org/pipermail/inquiry/2003-March/thread.html#196
ICE. http://stderr.org/pipermail/inquiry/2003-April/thread.html#363
hope this helps with the re*orientation ...
jon awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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